What you are seeing
Cut both disks along each other's boundary. Turning disk A by 1/n carries the arc of B's boundary that lies inside A to a new place; that arc is now a cut too, and some of it lies inside B, so turning B carries it further. Keep going until nothing new appears. What is left is every cut the puzzle can ever need, and the pieces are the regions between them.
For n = 2, 3, 4 and 6 the turns generate a lattice, and the cuts stay finite at every radius. For every other n there is a critical radius: below it the puzzle is finite, above it the cuts never stop and pile up into a fractal (Hearn, Kretschmer, Rokicki, Streeter and Vergo, Two-Disk Compound Symmetry Groups). The “critical radius” preset uses their table (n = 5 to 100); there the computation stops at the budget, so you see the fractal to a finite depth.
Turn the disks animates the puzzle. On a finite puzzle every turn maps the pieces onto pieces, so the cut pattern looks the same after each turn while the coloured pieces travel: pieces that start in one disk pass through the lens into the other. The piece count shown is traced live in your browser (every crossing, every face), independently of the table below.
Movement at any zoom. A turn carries a point about r·2π/n, which is far larger than the screen once you zoom in. So click a piece: the camera rides along with it through every turn, its disk turning with it while the rest of the puzzle swaps around it, and the slots it can ever reach are outlined. When a turn would sweep content across many screens, the page shows it as a crossfade instead: the cut pattern maps onto itself, so each slot simply takes on the colour of the piece that arrives there. The inset keeps track of which disk is turning and where you are.
The sequence
Once the disks overlap, a turn copies their shared lens n − 1 times around each disk, and the puzzle has 2n + 1 pieces. When neighbouring copies of the lens start to overlap, at r = sec(π/n), the count jumps and then stays constant over a wide range of r. The “plateau” preset picks a radius inside that range. The number of pieces there, for n = 3, 4, 5, …, is
For n ≥ 4 it fits a cubic quasi-polynomial of period 4, for example (n³ − n² + 4)/8 when n ≡ 2 (mod 4). These values are not yet in the OEIS.
How it is computed. Every cut is an arc of a circle of radius r whose centre lies in Z[ζn], so centres are stored as exact integer vectors and arc ends as “where this circle crosses that one”; nothing is rounded until the angles are drawn. The piece counts come from a separate Python program that also traces every piece and checks that their areas add up.